Complex analytic variety

In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.

Definition

Denote the constant sheaf on a topological space with value $\mathbb {C}$ by ${\underline {\mathbb {C} }}$. A $\mathbb {C}$-space is a locally ringed space $(X,{\mathcal {O}}_{X})$ whose structure sheaf is an algebra over ${\underline {\mathbb {C} }}$.

Choose an open subset $U$ of some complex affine space $\mathbb {C} ^{n}$, and fix finitely many holomorphic functions $f_{1},\dots ,f_{k}$ in $U$. Let $X=V(f_{1},\dots ,f_{k})$ be the common vanishing locus of these holomorphic functions, that is, $X=\{x\mid f_{1}(x)=\cdots =f_{k}(x)=0\}$. Define a sheaf of rings on $X$ by letting ${\mathcal {O}}_{X}$ be the restriction to $X$ of ${\mathcal {O}}_{U}/(f_{1},\ldots ,f_{k})$, where ${\mathcal {O}}_{U}$ is the sheaf of holomorphic functions on $U$. Then the locally ringed $\mathbb {C}$-space $(X,{\mathcal {O}}_{X})$ is a local model space.

A complex analytic variety is a locally ringed $\mathbb {C}$-space $(X,{\mathcal {O}}_{X})$ which is locally isomorphic to a local model space.

Morphisms of complex analytic varieties are defined to be morphisms of the underlying locally ringed spaces, they are also called holomorphic maps.

See also

• Analytic space
• Complex algebraic variety
• GAGA

References

• Grauert, Hans, and Reinhold Remmert. "Coherent analytic sheaves." Vol. 265. Springer Science & Business Media, 2012.
• Grauert, Peternell, and Remmert, Encyclopaedia of Mathematical Sciences 74: Several Complex Variables VII
• Cartan, H.; Bruhat, F.; Cerf, Jean.; Dolbeault, P.; Frenkel, Jean.; Hervé, Michel; Malatian.; Serre, J-P. "Séminaire Henri Cartan, Tome 4 (1951-1952)". (no.10-13)
• Serre, Jean-Pierre (1956), "Géométrie algébrique et géométrie analytique", Annales de l'Institut Fourier, 6: 1–42, doi:10.5802/aif.59, ISSN 0373-0956, MR 0082175
• Grothendieck, Alexander; Raynaud, Michele (2002). "Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique". Revêtements étales et groupe fondamental (SGA 1) (in French). arXiv:math/0206203. doi:10.1007/BFb0058656. ISBN 978-2-85629-141-2.
• . ISBN 978-4-431-49822-3. {{cite book}}: Missing or empty |title= (help)

External links

• Onishchik, A.L. (2001) [1994], "Analytic space", Encyclopedia of Mathematics, EMS Press
• Kiran Kedlaya. 18.726 Algebraic Geometry. Spring 2009. Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons BY-NC-SA.